500 ON THE REFLECTION OF LIGHT [422 and similarly the second factor may be written with change of sign of rj - {1 + AA'-t- T? (A + A')} {1 - AA'- ,7 (A'- A)}. Accordingly .Dll, a ._ 1(1 + * A? - ^2 ( A + A')*] {(1 - AAQ2 - ^ (A- AQ*} smn a 4^ A'2 (A2- I)2 ' "A ' In this, on restoring the values of A, A', 1 + A A' ± r, (A + A') = 2^<s+6') {cos ±k(S + 8')±rj cos ±k(S- 8')}, and 1 - A A' + 7? (A - A') = - 2^<W) {sin p (8 + S') + ?? sin £/<; (S - 8% Also 4 A'2 (A2 - 1)2 = - 16ett(S+s/) sin2 and thus • M _ lco sin n." tt — • „ • „ , 7 r> ?72 sin2 ^ KG x {sin2 p (5 + 8'} - tf sin2 i/c (S - 8')} ....... (55) The transition between the two cases (of opposite behaviour when m = oo) occurs when sinh a = 0. In general, this requires either 8') siniHS + 8') '. r\r r> = -\ -- - — - — ' - - ' ± conditions which are symmetrical with respect to 8 and 8', as clearly they ought to be*. In (55), (56), if is limited to values less than unity. Reverting to (43), we see that the evanescence of sinh2 « requires that r — ± 1 + t, or, if we separate the real and imaginary parts of r and t, r = ± 1 T <i + i<2. If, for example, we take r = — I — t, we have |r|2 = (l + 02 + ^=H-|«i2+2^. Also \r 2 = l-|i|2; so that r|2 = l + ^, t2 = -t1. In like manner by interchange of r and t, |/|2 — l-J-r /y.|2_ _ „, \ 1 1 — j. -t- T-L , r | — r1} showing that in this case r1} ^, are both negative. The general equation (55) shows that sinh2 a is negative, when rf- lies between sin2 This is the case (i) above denned where an increase in m leads to complete reflection. On the other hand, sinh2 a is positive when if lies outside the * That is -with reversal of the sign of 17, which makes uo difference here.unction of ?j. The first factor may be written